Frequency Domain Bootstrap

Abstract

In this chapter, we describe a special type of transformation based-bootstrap, known as the frequency domain bootstrap (FDB). Given a finite stretch of observations from a stationary time series, here we consider the discrete Fourier transforms (DFTs) of the data and use the transformed values in the frequency domain to derive bootstrap approximations (hence, the name FDB). In Section 9.2, we describe the FDB for a class of estimators, called the ratio statistics. Dahlhaus and Janas’s (1996) results show that under suitable regularity conditions, the FDB is second-order accurate for approximating the sampling distributions of ratio statistics. In Section 9.3, we describe the FDB method and its properties in the context of spectral density estimation. Material covered in Section 9.3 is based on the work of Franke and Hardle (1992). In Section 9.4, we describe a modified version of the FDB due to Kreiss and Paparoditis (2003) that, under suitable regularity conditions, removes some of the limitations of the standard FDB and yields valid approximations to the distributions of a larger class of statistics than the class of ratio statistics. It is worth pointing out that the results presented in this chapter on the FDB are valid only for linear processes.